Using AI to enhance customer service
This sounds to me like a use case for a chatbot. You would have different intents reflecting the types of user queries that your system can respond to. The intent matching can be done by pattern matching, machine learning (classification), or a combination of the two (hybrid). You can then use the chatbot to ask clarification questions or elicit more information to identify which live agent would be the best person to take over the call. Essentially each live agent would have a list of intents plus added information (such as geographical area etc) which you then compare against the data from the caller to find the best match.

If this is a voice call you'd need to put an ASR system at the front of the pipeline. Chatbots can usually do live-agent handover to then pass control to a human agent at any time in the conversation.

[Disclaimer: I work for a company that operates in exactly that area and whose system works as described above]